First Steps on the Representations of Domains (Extended
Abstract)

Marcelo P Fiore

Synopsis:Domain-theoretic categories are
axiomatised by means of categorical non-order-theoretic
requirements on a cartesian closed category equipped with a
dominance and a lifting. We show that every axiomatic
domain-theoretic category can be endowed with an intensional
notion of approximation, the path relation, with respect to
which the category Cpo-enriches. Subsequently, we provide a
representation theorem of the form: every small
domain-theoretic category D has a full and faithful
representation in Cpo[Dop, Set],
the category of cpos and continuous functions in the presheaf topos
[Dop, Set]. Our analysis suggests more
liberal notions of domains. In particular, we present a category
where the path preorder is not omega-complete, but in which
the constructions of domain theory (as, for example, the existence
of uniform fixed-point operators and the solution of domain
equations) are possible.